Is it possible to have 2 different individuals responsible for the same task?

Of course it is. It happens all the time. It has the benefit to management that they can pick and choose which one (or both) to sack when it goes wrong. Of course it means that people have to send out two emails when enquiring about progress or reporting back on work done.

I'm guessing you mean two instances of the same task?_________________"Method goes far to prevent trouble in business: for it makes the task easy, hinders confusion, saves abundance of time, and instructs those that have business depending, both what to do and what to hope."
William Penn 1644-1718

Is it possible to have 2 different individuals responsible for the same task?

Of course it is. It happens all the time. It has the benefit to management that they can pick and choose which one (or both) to sack when it goes wrong. Of course it means that people have to send out two emails when enquiring about progress or reporting back on work done.

I'm guessing you mean two instances of the same task?

Same instance same task. ie can the problem investigator and the problem assignee both be responsible to deliver the required work needed?

People can only be responsible for the work they are required to do and to whomsoever legitimately requires it of them. So the investigator cannot be responsible for the work the assignee does, unless the assignee is doing it for the investigator and then the assignee is only responsible to the investigator. And even then only if that is a legitimate commissioning/supervisory relationship.

Why would you want confusion?_________________"Method goes far to prevent trouble in business: for it makes the task easy, hinders confusion, saves abundance of time, and instructs those that have business depending, both what to do and what to hope."
William Penn 1644-1718

Typical steps in a RACI process:
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1. Identify all of the processes / activities involved and list them down the left hand side of the chart.
2. Identify all of the roles and list them along the top of the chart.
3. Complete the cells of the chart: identify who has the R, A, S, C, I for each process.
4. Every process should preferably have one and only one “R” as a general principle. A gap occurs when a process exists with no “R” (no role is responsible), an overlap occurs when multiple roles exist that have an “R” for a given process.
5. Resolve Overlaps - Every process in a role responsibility map should contain one and only one “R” to indicate a unique process owner. In the case of multiple “R”s, there is a need to “zoom in” and further detail the sub processes associated with “obtain resource commitment” to separate the individual responsibilities.
6. Resolve Gaps - The simpler case to address is the resolution of a gap. Where no role is identified that is “responsible” for a process, the individual with the authority for role definition must determine which existing role is responsible or new role that is required, update the RASCI map and clarify with the individual(s) that assume that role.

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I've completed a RACI for problem management and would welcome sharing with others to see comparisons etc.